A smoke plume that has become nearly horizontal at some distance from the source behaves much like a ‘line-thermal’, for which, using a perturbation method, a solution in laminar flow may be obtained, on the supposition that excess temperatures are small and buoyant movements slow, i.e. that the Rayleigh number of the problem is suitably low. In analogy with some other problems in turbulent flow and turbulent diffusion, the laminar solution is then assumed to approximate what is observed in the turbulent case, provided that the rate of growth of the diffusing cloud is assessed realistically.

The so-calculated pattern of streamlines in a cross-section of the plume agrees qualitatively with the observed behaviour of hot plumes and puffs, consisting of two vortex-like structures of opposite sense of rotation, lying on either side of the plume centre. The bodily upward movement of the plume is found to depend critically on the rate of growth of the plume. Thus when the plume diameter grows faster than linearly with distance (such behaviour characterizes the ‘quasi-asymptotic’ stage of relative diffusion predicted by Batchelor, 1952, for which Richardson's (4/3)-power law of eddy diffusivity holds) the plume tends to reach an asymptotic height. A crude theoretical estimate of the asymptotic height attained shows fair agreement with observations reported elsewhere. Although the plume nearly reaches this asymptotic height in the quasi-asymptotic phase, it retains a small gradient in the final phase which may be of importance at large distances from the source. The small-Rayleigh-number criterion restricts the validity of the solution to ‘weakly buoyant’ plumes.